A 150 km wide ground deformation anomaly in the Altiplano-Puna Volcanic Complex (APVC), with uplift centered on Uturuncu volcano and peripheral subsidence, alludes to complex subsurface stress changes. In particular, the role of a large, geophysically anomalous and partially molten reservoir (the Altiplano-Puna Magma Body APMB) located ~20 km beneath the deforming surface, is still poorly understood. To explain the observed spatio-temporal ground deformation pattern, we integrate geophysical and petrological data and develop a numerical model that accounts for a mechanically heterogeneous and visco-elastic crust. Best-fit models imply subsurface stress changes due to the episodic reorganization of an interconnected vertically-extended mid-crustal plumbing system composed of the APMB and a domed bulge-and-column structure. Measured gravity-height gradients data point towards low-density fluid migration as the dominant process behind the stress changes. We calculate a mean annual flux of ~2*107 m3 of water-rich andesitic melt and/or magmatic water from the APMB into the bulge-and-column accompanied by modest pressure changes of <0.006 MPa/year. Two configurations of the column fit the observations equally well: i) a magmatic (igneous mush) column that extends to a depth of 6 km b.s.l and contains trapped volatiles or ii) a volatile-bearing hybrid column composed of an igneous mush below a solidified and permeable body that extends to sea level. Volatile loss from the bulge-and-column reverses the deformation, and explains the absence of broad (tens of km) and long-term (>100 years) residual deformation at Uturuncu. Episodic mush reorganization may be an ubiquitous characteristic of the magmatic evolution of the APVC.